Saturday, July 20, 2013

Bernhard Riemann: an anniversary

Georg Friedrich Bernhard Riemann was born in a village in the Kingdom of Hanover on September 17th, 1826 and died in Selasca (Verbania), Northern Italy, on July 20th, 1866, i.e. 147 years ago. As you can see, he was 2 months short of 40 years when he died.

His father was a poor Lutheran pastor and a Napoleonic war veteran. His mother died when he was a kid. He was the second (oldest) among six children. Bernhard was shy, timid, afraid of speaking in public, and suffered from psychological problems and breakdowns. His math skills were obvious early on. However, he started to study a lyceum and investigated the Bible intensely, with a rough plan to become a pastor to earn some money for his family.

In 1846, his father gathered enough money to send Bernhard to University of Göttingen – to study... theology. But this has always been quite a place for mathematics. It didn't take much time for Riemann to study under Carl Friedrich Gauss himself, especially the method of least squares and similar things. After an approval from his father, Bernhard moved to University of Berlin where people like Jacobi, Dirichlet, Steiner, and Eisenstein were just teaching. After two years, in 1849, he returned to Göttingen.

In 1854, he was giving his first lectures – at the age of 28 – and the topic was nothing else than... the Riemannian geometry. These were times when without much ado, results of this magnitude could have been presented as an ordinary lecture for students. Some "extraordinary professor Riemann" efforts failed but at least, he started to get a proper salary and after Dirichlet's death, he would become the head of the mathematics department.

He married Elise Koch and they had a daughter. Riemann wanted to avoid wars so during the Austro-Prussian war that was taking place in Göttingen, he began to flee to Italy. Tuberculosis killed him there in July 1866, 34 days before that so-called Seven Weeks' War ended by Peace of Prague – Prussia gained some territory and my empire of Austria lost its previous right to define the intra-German relations and unification (the latter effectively started from scratch). His health has always been unreliable.

Riemann was a pure mathematician and among pure mathematicians, he was one of the most influential ones when it comes to the impact on physics. He not only invented the non-Euclidean (or generalized Euclidean) Riemannian geometry that became the basis of Einstein's general theory of relativity (substitute a long essay on the liberation out of the Euclidean straitjacket here). He was also the first man to propose that the physical reality could have extra dimensions (65 years before Kaluza initiated the research of Kaluza-Klein theory), something that only became obvious with the rise of string theory a few decades ago (about 120 years after Riemann's lecture). Many of the things we are using today were presented by Riemann in 1854 in their final form. Many others are more modern and would undoubtedly make Riemann happy.

His contributions to mathematics – the kind of mathematics that is mostly important for physicists (and the type of mathematics that you often encounter on this blog) – were numerous. Riemann surfaces, their topology, Riemann integrals using Riemann sums, Riemann–Liouville differintegral, Riemann zeta function, Riemann hypothesis, monodromies and the hypergeometric functions, and so on, and so on. It's really remarkable how many deep discoveries this former would-be theologian has done in less than 40 years that a mingy God gave him to spend on Earth.

9 comments:

Riemann was, of course, above all a mathematician but he was more directly interested in physics than you make it seem in this account. He wrote papers on the theory of gases, fluid dynamics, heat, light, magnetism and electricity and acoustics. For his famous habilitation lecture he proposed three topics: two of them were on topics related to electricity and magnetism. Actually he was working on “unified field theory” - (as he says in his own words: “the connection between electricity, magnetism, light and gravitation”) a subject that occupied him to the end of his life. It was to everyone’s surprise that Gauss chose the third topic: which turned out to be what we know call riemannian geometry. But that lecture was as much philosophical as mathematical and he considered not only the abstract mathematical notion of “space” but also speculated about the nature of physical space.

It seems to me that one can make a pretty good case for viewing him as a pre-founder of String Theory.

You had to be mightily impressive to get high praise from Gauss because he often infuriated other mathematicians when they announced outstanding results only to have Gauss airily say that he had the same thoughts years ago. Gauss was effusive about Riemann's thesis and his habilitation lecture.